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• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

INTRODUCTION TO COMPUTATIONAL INTELLIGENCE

Lin Shang Dept. of Computer Science and Technology

Nanjing University

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

关于课程：CI

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Fuzzy Set and Rough Set

n Introduction n History and definition

n Fuzzy Sets n Membership function n Fuzzy set operations

n Rough Sets n Approximation n Reduction

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

“Fuzzification is a kind of scientific permisiveness; it tends to result in socially appealing slogans unaccompanied by the discipline of hard work.”

R. E. Kalman, 1972

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Set

Fuzzy Set

Rough Set

(collections) of various objects of interest

“Number of things of the same kind, that belong together because they are similar or complementary to each other.” The Oxford English Dictionary

Set Theory: George Cantor (1893)

an element can belong to a set to a degree k (0 ≤ k ≤ 1)

completely new, elegant approach to vagueness

Fuzzy Set theory: Lotfi Zadeh(1965)

imprecision is expressed by a boundary region of a set

another approach to vagueness

Rough Set Theory: Zdzisaw Pawlak(1982)

Introduction

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Introduction

n Early computer science • Not good at solving real problems • The computer was unable to make accurate inferences • Could not tell what would happen, give some

preconditions • Computer always seemed to need more information

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

n “Fuzzy Sets” paper published in 1965 n Comprehensive - contains everything needed to implement

FL n Key concept is that of membership values: extent to which an object meets vague or imprecise properties n Membership function: membership values over domain of

interest n Fuzzy set operations n Awarded the IEEE Medal of Honor in 1995

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

History for fuzzy sets and system

n First fuzzy control system, work done in 1973 with Assilian (1975)

n Developed for boiler-engine steam plant

n 24 fuzzy rules

n Developed in a few days

n Laboratory-based

n Served as proof-of-concept

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Early European Researchers

Hans Zimmerman, Univ. of Aachen •Founded first European FL working group in 1975 •First Editor of Fuzzy Sets and Systems •First President of Int’l. Fuzzy Systems Association

Didier Dubois and Henri Prade in France •Charter members of European working group •Developed families of operators •Co-authored a textbook (1980)

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Early U. S. Researchers

K. S. Fu (Purdue) and Azriel Rosenfeld (U. Md.) (1965-75)

Enrique Ruspini at SRI •Theoretical FL foundations •Developed fuzzy clustering

James Bezdek, Univ. of West Florida •Developed fuzzy pattern recognition algorithms •Proved fuzzy c-means clustering algorithm •Combined fuzzy logic and neural networks •Chaired 1st Fuzz/IEEE Conf. in 1992 and others •President of IEEE NNC 1997-1999

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

The Dark Age

•Lasted most of 1980s

•Funding dried up, in US especially

“...Fuzzy logic is based on fuzzy thinking. It fails to distinguish between the issues specifically addressed by the traditional methods of logic, definition and statistical decision- making...”

- J. Konieki (1991) in AI Expert

•Symbolics ruled: “fuzzy” label amounted to the ‘kiss of death’

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Michio Sugeno

•Secretary of Terano’s FL working group, est. in 1972

•1974 Ph.D. dissertation: fuzzy measures theory

•Worked in UK

•First commercial application of FL in Japan: control system for water purification plant (1983)

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Other Japanese Developements n 1st consumer product: shower head using FL circuitry to

control temperature (1987)

n Fuzzy control system for Sendai subway (1987)

n 2d annual IFSA conference in Tokyo was turning point for FL (1987)

n Laboratory for Int’l. Fuzzy Engineering Research (LIFE) founded in Yokohama with Terano as Director, Sugeno as Leading Advisor in 1989.

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Fuzzy Systems Theory and Paradigms

n Variation on 2-valued logic that makes analysis and control of real (non-linear) systems possible

n Crisp “first order” logic is insufficient for many applications because almost all human reasoning is imprecise

n fuzzy sets, approximate reasoning, and fuzzy logic issues and applications

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Fuzziness is not probability

• Probability is used, for example, in weather forecasting

• Probability is a number between 0 and 1 that is the

certainty that anevent will occur

• The event occurrence is usually 0 or 1 in crisp logic, but

fuzziness says that it happens to some degree

• Fuzziness is more than probability; probability is a subset

of fuzziness

• Probability is only valid for future/unknown events

• Fuzzy set membership continues after the event

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Probability

• Probability is based on a closed world model in which it is assumed that everything is known

• Probability is based on frequency; Bayesian on subjectivity

• Probability requires independence of variables

• In probability, absence of a fact implies knowledge

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Fuzzy Sets

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Fuzzy Set Membership

•In fuzzy logic, set membership occurs by degree •Set membership values are between 0 and 1 •We can now reason by degree, and apply logical operations to fuzzy sets

We usually write

or, the membership value of x in the fuzzy set A is m, where

mxA =)(µ

10 ≤≤ m

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Fuzzy Set Membership Functions

• Fuzzy sets have “shapes”: the membership values plotted versus the variable

• Fuzzy membership function: the shape of the fuzzy set over the range of the numeric variable Can be any shape, including arbitrary or irregular Is normalized to values between 0 and 1 Often uses triangular approximations to save computation time

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Fuzzy Sets Are Membership Functions

from Bezdek

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Representations of Membership Functions

⎭ ⎬ ⎫

⎩ ⎨ ⎧ ++++=

⎭ ⎬ ⎫

⎩ ⎨ ⎧ ++=

90 0

80 5.

70 1

60 5.

50 0

15.2 1

95.1 50.

75.1 0

Warm

TAMPBP

( ) ( ) 50/80_ 2−−= pPRICEFAIR epµ

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Two Types of Fuzzy Membership Function

• INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2016

Equality of Fuzzy Sets

• In traditional logic, sets containing the same members are equal: {A,B,C} = {A,B,C}

• In fuzzy logic, however, two sets are equal if and only if all elements have identical membership values:

{.1/

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